SPARS trial A
Stimulus-response characteristics of the SPARS
Peter Kamerman
10 Feb 2018
This analysis examines the stimulus-response characateristics of the SPARS.
Import and inspect data
# Import
data <- read_rds('./data-cleaned/SPARS_A.rds')
# Inspect
glimpse(data)## Observations: 1,927
## Variables: 19
## $ PID <chr> "ID01", "ID01", "ID01", "ID01", "ID01", "ID0...
## $ block <chr> "A", "A", "A", "A", "A", "A", "A", "A", "A",...
## $ block_order <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
## $ trial_number <dbl> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1...
## $ intensity <dbl> 3.75, 1.50, 3.25, 1.50, 3.00, 2.75, 1.00, 2....
## $ intensity_char <chr> "3.75", "1.50", "3.25", "1.50", "3.00", "2.7...
## $ rating <dbl> -10, -40, -10, -25, -20, -25, -40, 2, -40, -...
## $ rating_positive <dbl> 40, 10, 40, 25, 30, 25, 10, 52, 10, 40, 54, ...
## $ EDA <dbl> 18315.239, 13904.177, 11543.449, 20542.834, ...
## $ age <dbl> 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, ...
## $ sex <dbl> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,...
## $ panas_positive <dbl> 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, ...
## $ panas_negative <dbl> 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, ...
## $ dass42_depression <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
## $ dass42_anxiety <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
## $ dass42_stress <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
## $ pcs_magnification <dbl> 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,...
## $ pcs_rumination <dbl> 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, ...
## $ pcs_helplessness <dbl> 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, ...Clean and process data
We performed a basic clean-up of the data, and then calculated Tukey trimean at each stimulus intensity for each participant (participant average), and finally the median of the trimeans at each stimulus intensity across participants (group average).
############################################################
# #
# Clean #
# #
############################################################
data %<>%
# Select required columns
select(PID, block, block_order, trial_number, intensity, intensity_char, rating)
############################################################
# #
# Calculate 'Tukey trimean' #
# #
############################################################
# Define tri.mean function
tri.mean <- function(x) {
# Calculate quantiles
q1 <- quantile(x, probs = 0.25, na.rm = TRUE)[[1]]
q2 <- median(x, na.rm = TRUE)
q3 <- quantile(x, probs = 0.75, na.rm = TRUE)[[1]]
# Calculate trimean
tm <- (q2 + ((q1 + q3) / 2)) / 2
# Convert to integer
tm <- as.integer(round(tm))
return(tm)
}
# Calculate the participant average
data_tm <- data %>%
group_by(PID, intensity) %>%
summarise(tri_mean = tri.mean(rating)) %>%
ungroup()
# Calculate the group average
data_group <- data_tm %>%
group_by(intensity) %>%
summarise(median = median(tri_mean)) %>%
ungroup()Exploratory plots
Group-level stimulus response curve
# Plot
data_tm %>%
ggplot(data = .) +
aes(x = intensity,
y = tri_mean) +
geom_point(position = position_jitter(width = 0.05)) +
geom_smooth(method = 'loess',
se = FALSE,
colour = '#666666',
size = 0.6) +
geom_point(data = data_group,
aes(y = median),
shape = 21,
size = 4,
fill = '#D55E00') +
labs(title = 'Group-level stimulus-response plots',
subtitle = 'Black circles: participant-level Tukey trimeans | Orange circles: group-level median | Grey line: loess curve',
x = 'Stimulus intensity (J)',
y = 'SPARS rating [-50 to 50]') +
scale_y_continuous(limits = c(-50, 50)) +
scale_x_continuous(breaks = seq(from = 1, to = 4, by = 0.5))
Participant-level stimulus response curves
All trials
# Plot
data %>%
ggplot(data = .) +
aes(x = intensity,
y = rating) +
geom_point() +
geom_smooth(method = 'loess',
se = FALSE,
colour = '#666666',
size = 0.6) +
geom_point(data = data_tm,
aes(y = tri_mean),
shape = 21,
size = 3,
fill = '#D55E00') +
labs(title = 'Participant-level stimulus-response plot',
subtitle = 'Black circles: individual experimental blocks | Orange circles: Tukey trimean | Grey line: loess curve',
x = 'Stimulus intensity (J)',
y = 'SPARS rating [-50 to 50]') +
scale_y_continuous(limits = c(-50, 50)) +
facet_wrap(~ PID, ncol = 4)
Trials by experimental block
# Process data
data_block <- data %>%
# Rename blocks
mutate(block = sprintf('Block: %s (order: %i)', block, block_order)) %>%
# Nest by PID
group_by(PID) %>%
nest() %>%
# Generate plots
mutate(plots = map2(.x = data,
.y = unique(PID),
~ ggplot(data = .x) +
aes(x = intensity,
y = rating) +
geom_point() +
geom_smooth(method = 'loess',
se = FALSE,
colour = '#666666',
size = 0.6) +
labs(title = paste(.y, ': Participant-level stimulus-response plots conditioned on experimental block'),
subtitle = 'Black circles: individual data points | Grey line: loess curve',
x = 'Stimulus intensity (J)',
y = 'SPARS rating [-50 to 50]') +
scale_y_continuous(limits = c(-50, 50)) +
facet_wrap(~ block, ncol = 2)))
# Print plots
walk(.x = data_block$plots, ~ print(.x))
Linear mixed model regression
To allow for a curvilinear relationship between stimulus intensity and rating, we modelled the data using polynomial regression, with 1st (linear), 2nd (quadratic), and 3rd (cubic) order orthogonal polynomials. For each polynomial expression, we modelled the random effects as random intercept only, and as random intercept and slope.
The random intercept only and random intercept and slope models were compared using the logliklihood test, and the better model taken foward. Diagnostics were run on the final model only, and we examined level 1 residuals (conditional / fixed effects), and level 2 residuals (random effects) and influence points 1.
1st-order (linear) polynomial
# Intercept only
lmm1 <- lmer(tri_mean ~ intensity + (1 | PID),
data = data_tm,
REML = TRUE)
# Intercept and slope
lmm1b <- lmer(tri_mean ~ intensity + (intensity | PID),
data = data_tm,
REML = TRUE)
# Better model?
anova(lmm1, lmm1b)## Data: data_tm
## Models:
## lmm1: tri_mean ~ intensity + (1 | PID)
## lmm1b: tri_mean ~ intensity + (intensity | PID)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## lmm1 4 1814.7 1828.7 -903.37 1806.7
## lmm1b 6 1733.6 1754.6 -860.79 1721.6 85.146 2 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Anova of better model
Anova(lmm1b,
type = 2,
test.statistic = 'F')## Analysis of Deviance Table (Type II Wald F tests with Kenward-Roger df)
##
## Response: tri_mean
## F Df Df.res Pr(>F)
## intensity 94.707 1 17.998 1.356e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Print better model
sjt.lmer(lmm1b,
show.header = TRUE,
string.dv = "Response",
string.pred = "Coefficients",
depvar.labels = '',
pred.labels = 'intensity',
string.est = 'Estimate',
string.ci = '95% CI',
string.p = 'p-value',
show.icc = FALSE,
show.r2 = FALSE)| Coefficients | Response | |||
| Estimate | 95% CI | p-value | ||
| Fixed Parts | ||||
| (Intercept) | -39.76 | -51.32 – -28.21 | <.001 | |
| intensity | 14.13 | 11.28 – 16.97 | <.001 | |
| Random Parts | ||||
| σ2 | 42.542 | |||
| τ00, PID | 633.161 | |||
| ρ01 | -0.887 | |||
| NPID | 19 | |||
| Observations | 244 | |||
2nd-order (quadratic) polynomial
# Intercept only
lmm2 <- lmer(tri_mean ~ poly(intensity, 2) + (1 | PID),
data = data_tm,
REML = TRUE)
# Intercept and slope
lmm2b <- lmer(tri_mean ~ poly(intensity, 2) + (intensity | PID),
data = data_tm,
REML = TRUE)
# Better model?
anova(lmm2, lmm2b)## Data: data_tm
## Models:
## lmm2: tri_mean ~ poly(intensity, 2) + (1 | PID)
## lmm2b: tri_mean ~ poly(intensity, 2) + (intensity | PID)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## lmm2 5 1816.7 1834.2 -903.35 1806.7
## lmm2b 7 1735.5 1760.0 -860.74 1721.5 85.22 2 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Anova for better model
Anova(lmm2b,
type = 2,
test.statistic = 'F')## Analysis of Deviance Table (Type II Wald F tests with Kenward-Roger df)
##
## Response: tri_mean
## F Df Df.res Pr(>F)
## poly(intensity, 2) 46.667 2 43.413 1.526e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Print better model
sjt.lmer(lmm2b,
show.header = TRUE,
string.dv = "Response",
string.pred = "Coefficients",
depvar.labels = '',
pred.labels = 'intensity',
string.est = 'Estimate',
string.ci = '95% CI',
string.p = 'p-value',
show.icc = FALSE,
show.r2 = FALSE)| Coefficients | Response | |||
| Estimate | 95% CI | p-value | ||
| Fixed Parts | ||||
| (Intercept) | -4.67 | -10.91 – 1.57 | .160 | |
| poly(intensity, 2)1 | 205.33 | 163.97 – 246.69 | <.001 | |
| poly(intensity, 2)2 | 2.06 | -10.78 – 14.91 | .757 | |
| Random Parts | ||||
| σ2 | 42.727 | |||
| τ00, PID | 633.218 | |||
| ρ01 | -0.887 | |||
| NPID | 19 | |||
| Observations | 244 | |||
3rd-order (cubic) polynomial
# Intercept only
lmm3 <- lmer(tri_mean ~ poly(intensity, 3) + (1 | PID),
data = data_tm,
REML = TRUE)
# Intercept and slope
lmm3b <- lmer(tri_mean ~ poly(intensity, 3) + (intensity | PID),
data = data_tm,
REML = TRUE)
# Better model?
anova(lmm3, lmm3b)## Data: data_tm
## Models:
## lmm3: tri_mean ~ poly(intensity, 3) + (1 | PID)
## lmm3b: tri_mean ~ poly(intensity, 3) + (intensity | PID)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## lmm3 6 1813.8 1834.8 -900.90 1801.8
## lmm3b 8 1727.0 1754.9 -855.48 1711.0 90.841 2 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Anova for better model
Anova(lmm3b,
type = 2,
test.statistic = 'F')## Analysis of Deviance Table (Type II Wald F tests with Kenward-Roger df)
##
## Response: tri_mean
## F Df Df.res Pr(>F)
## poly(intensity, 3) 34.148 3 71.491 8.318e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Print better model
sjt.lmer(lmm3b,
show.header = TRUE,
string.dv = "Response",
string.pred = "Coefficients",
depvar.labels = '',
pred.labels = 'intensity',
string.est = 'Estimate',
string.ci = '95% CI',
string.p = 'p-value',
show.icc = FALSE,
show.r2 = FALSE)| Coefficients | Response | |||
| Estimate | 95% CI | p-value | ||
| Fixed Parts | ||||
| (Intercept) | -4.67 | -10.89 – 1.56 | .159 | |
| poly(intensity, 3)1 | 205.35 | 163.69 – 247.01 | <.001 | |
| poly(intensity, 3)2 | 2.12 | -10.42 – 14.67 | .744 | |
| poly(intensity, 3)3 | 20.95 | 8.40 – 33.49 | .004 | |
| Random Parts | ||||
| σ2 | 40.768 | |||
| τ00, PID | 639.311 | |||
| ρ01 | -0.889 | |||
| NPID | 19 | |||
| Observations | 244 | |||
Compare models
knitr::kable(broom::tidy(anova(lmm1b, lmm2b, lmm3b)),
caption = 'Linear model vs quadratic model and cubic model')| term | df | AIC | BIC | logLik | deviance | statistic | Chi.Df | p.value |
|---|---|---|---|---|---|---|---|---|
| lmm1b | 6 | 1733.586 | 1754.569 | -860.7930 | 1721.586 | NA | NA | NA |
| lmm2b | 7 | 1735.487 | 1759.967 | -860.7434 | 1721.487 | 0.0991866 | 1 | 0.7528079 |
| lmm3b | 8 | 1726.958 | 1754.936 | -855.4791 | 1710.958 | 10.5285980 | 1 | 0.0011754 |
PLot the model
predicted <- ggeffects::ggpredict(model = lmm3b,
terms = 'intensity',
ci.lvl = 0.95)
ggplot() +
geom_ribbon(data = predicted,
aes(x = x,
ymin = conf.low,
ymax = conf.high),
fill = '#cccccc') +
geom_line(data = predicted,
aes(x = x,
y = predicted)) +
geom_point(data = predicted,
aes(x = x,
y = predicted)) +
geom_point(data = data_group,
aes(x = intensity,
y = median),
shape = 21,
size = 4,
fill = '#D55E00') +
labs(title = 'Cubic model (95% CI): Predicted values vs stimulus intensity',
subtitle = 'Black circles/line: predicted values | Orange circles: group-level median',
x = 'Stimulus intensity (J)',
y = 'SPARS rating [-50 to 50]') +
scale_y_continuous(limits = c(-50, 50)) +
scale_x_continuous(breaks = seq(from = 1, to = 4, by = 0.5))
The cubic model has the best fit. The resulting curvilinear response function is steepest at the extremes and flattens out in the mid-ranges of stumulus intensity. We performed diagnostics on this model to confirm that the model was properly specified.
Diagnostics on the cubic model
Generate residuals
# Level 1 residuals
## Standardized
lmm_resid1 <- HLMresid(lmm3b,
level = 1,
type = 'LS',
standardize = TRUE)
# Semi-standardized residuals (used for assessing homoscedasticity)
lmm_ssresid1 <- HLMresid(lmm3b,
level = 1,
type = 'LS',
standardize = 'semi')
# Level 2 residuals
## Standardized
lmm_resid2 <- HLMresid(lmm3b,
level = 'PID',
type = 'EB') Level 1 residuals: linearity
The relationship between predictor(s) and outcome for a linear model should be linear. This relationship can be observed by plotting the level 1 standardized residuals against the predictors. The scatter of residuals should show no pattern, and be centered around 0.
# Standardized residuals vs intensity
ggplot(data = lmm_resid1) +
aes(x = `poly(intensity, 3)`[, 1],
y = std.resid) +
geom_point() +
geom_smooth(method = 'lm') +
geom_hline(yintercept = 0) +
geom_hline(yintercept = -2,
linetype = 2) +
geom_hline(yintercept = 2,
linetype = 2) +
labs(title = 'Cubic model: Level 1 residuals vs intensity',
subtitle = 'Assess linearity of the intensity term | Blue line: linear regression line',
caption = 'The regression line should be centered on 0\n~95% of points should be betwen -2 and +2',
y = 'Standardized residuals',
x = 'Stimulus intensity')
# Standardized residuals vs intensity^2
ggplot(data = lmm_resid1) +
aes(x = `poly(intensity, 3)`[, 2],
y = std.resid) +
geom_point() +
geom_smooth(method = 'lm') +
geom_hline(yintercept = 0) +
geom_hline(yintercept = -2,
linetype = 2) +
geom_hline(yintercept = 2,
linetype = 2) +
labs(title = expression(paste('Cubic model: Level 1 residuals vs ', intensity^2)),
subtitle = expression(paste('Assess linearity of the ', intensity^2, ' term | Blue line: linear regression line')),
caption = 'The regression line should be centered on 0\n~95% of points should be betwen -2 and +2',
y = 'Standardized residuals',
x = expression(Stimulus~intensity^2)) 
# Standardized residuals vs intensity^3
ggplot(data = lmm_resid1) +
aes(x = `poly(intensity, 3)`[, 3],
y = std.resid) +
geom_point() +
geom_smooth(method = 'lm') +
geom_hline(yintercept = 0) +
geom_hline(yintercept = -2,
linetype = 2) +
geom_hline(yintercept = 2,
linetype = 2) +
labs(title = expression(paste('Cubic model: Level 1 residuals vs ', intensity^3)),
subtitle = expression(paste('Assess linearity of the ', intensity^3, ' term | Blue line: linear regression line')),
caption = 'The regression line should be centered on 0\n~95% of points should be betwen -2 and +2',
y = 'Standardized residuals',
x = expression(Stimulus~intensity^3)) 
The regression curve for the quadratic term shows some signs of deviating from slope = 0, but otherwise the model specification (in terms of linearity) looks okay. Based on the overall pictire, we accept that the condition of linearity for the cubic model.
Level 1 residuals: homoscedasticity
The variance of residuals should be constant across the range of the predictor(s). This relationship can be observed by plotting the level 1 semi-standardized residuals against the predictors. Like the assessment of linearity, the residuals should be centered on 0, and show no pattern in the scatter of points.
# Standardized residuals vs intensity
ggplot(data = lmm_ssresid1) +
aes(x = `poly(intensity, 3)`[, 1],
y = semi.std.resid) +
geom_point() +
geom_hline(yintercept = 0) +
labs(title = 'Cubic model: Level 1 residuals vs intensity',
subtitle = 'Assess homoscedasticity for the intensity term',
y = 'Semi-standardized residuals',
x = 'Stimulus intensity')
# Standardized residuals vs intensity^2
ggplot(data = lmm_ssresid1) +
aes(x = `poly(intensity, 3)`[, 2],
y = semi.std.resid) +
geom_point() +
geom_hline(yintercept = 0) +
labs(title = expression(paste('Cubic model: Level 1 residuals vs ', intensity^2)),
subtitle = expression(paste('Assess homoscedasticity for the ', intensity^2, ' term')),
y = 'Semi-standardized residuals',
x = expression(Stimulus~intensity^2)) 
# Standardized residuals vs intensity^3
ggplot(data = lmm_ssresid1) +
aes(x = `poly(intensity, 3)`[, 3],
y = semi.std.resid) +
geom_point() +
geom_hline(yintercept = 0) +
labs(title = expression(paste('Cubic model: Level 1 residuals vs ', intensity^3)),
subtitle = expression(paste('Assess homoscedasticity for the ', intensity^3, ' term')),
y = 'Semi-standardized residuals',
x = expression(Stimulus~intensity^3)) 
There is no obvious pattern to the scatter of residuals across any of the fixed effect terms. So we accept that the residuals are homoscedastic in the cubic model.
Level 1 residuals: residual distribution
Residuals should be normally distributed. There are various methods of examining the distribution, and we have chosen the QQ-plot method, which plots the quantiles of the standardized residuals against a theoretical (Gaussian) quantile distribution. Points should line on the line of identity of the two sets of quantiles follow the same distribution.
# Standardized residuals vs intensity
ggplot_qqnorm(x = lmm_resid1$std.resid,
line = "rlm") +
labs(title = 'Cubic model: QQ-plot of level 1 residuals',
subtitle = 'Assessing whether residuals follow a normal distribution',
x = 'Theoretical quantiles',
y = 'Standardized residuals')
There is minor deviation at the extremes, but on the whole, we are satisfied that the cubic model fits the assumption of normally sdistributed residuals.
Level 2 residuals: residual distribution
Level 2 residuals can be used to identify predictors that should be included in the model, but since we are only assessing the effect of stimulus strength on SPARS rating, we have only assessed whether the level 2 residuals (intercept and slope) meet the assumption of being normally distributed (assessed using QQ-plots).
# Generate QQplots
qq1 <- ggplot_qqnorm(x = lmm_resid2$`(Intercept)`,
line = "rlm") +
labs(title = 'Cubic model: QQ-plot of level 2 residuals (Intercept)',
subtitle = 'Assessing whether residuals follow a normal distribution',
x = 'Theoretical quantiles',
y = 'Residuals')
qq2 <- ggplot_qqnorm(x = lmm_resid2$intensity,
line = "rlm") +
labs(title = 'Cubic model: QQ-plot of level 2 residuals (slope: intensity)',
subtitle = 'Assessing whether residuals follow a normal distribution',
x = 'Theoretical quantiles',
y = 'Residuals')
# Plot
qq1 + qq2
Although the data are sparse, we are satisfied that the level 2 residuals for the intercept and the slope of the cubic model fit the assumption of being normally sdistributed.
influence points
We assessed three aspects of influence (data that significantly model coefficients):
The variance component (random effects) was assesed using the relative variance change metric, which calculates the impact of deleting observational units of the variance of the residuals, random intercept, random slope, and covariance of the random slope and random intercept.
Leverage was used to assess fitted values. The assessment involves assessing the rate of change in the predicted response with respect to the observed response.
Cook’s Distance was used to assess the influence of fixed effects. The metric measures the distance between the fixed effects estimates obtained from the full model to that obtained from the reduced data (observations removed).
In all cases, we treated the individual (indicated using PID) as the unit of observation, and we used internal scaling to set the diagnostic cutoffs for each metric. The cutoffs were determined as: \(3^{rd}~Quartile + (3 \cdot IQR)\).
# Prepare relative variance change (RCV)
influence_rvc <- rvc(lmm3b,
group = 'PID')
# Prepare Cook's distance
influence_cooks <- cooks.distance(lmm3b,
group = 'PID')
# Prepare leverage
## (Assessed at the level of PID, and not the individual observation)
influence_leverage <- leverage(lmm3b,
level = 'PID')Random effects
Estimation of the variance component was undertaken by calculating relative variance change (RCV). RVC is close to zero when deletion of observational units from the model does not have a large infuence on the variance component.
# Plot
dotplot_diag(x = influence_rvc[ , 1],
cutoff = 'internal',
name = 'rvc') +
labs(title = 'Relative variance change for the residual variance',
subtitle = 'Cutoffs determined by measures of internal scaling',
y = 'Relative variance change',
x = 'Participant ID')
dotplot_diag(x = influence_rvc[ , 2],
cutoff = 'internal',
name = 'rvc') +
labs(title = 'Relative variance change for the random intercept variance',
subtitle = 'Cutoffs determined by measures of internal scaling',
y = 'Relative variance change',
x = 'Participant ID')
dotplot_diag(x = influence_rvc[ , 3],
cutoff = 'internal',
name = 'rvc') +
labs(title = 'Relative variance change for the random slope variance',
subtitle = 'Cutoffs determined by measures of internal scaling',
y = 'Relative variance change',
x = 'Participant ID')
dotplot_diag(x = influence_rvc[ , 4],
cutoff = 'internal',
name = 'rvc') +
labs(title = 'Relative variance change for the random slope and intercept covariance',
subtitle = 'Cutoffs determined by measures of internal scaling',
y = 'Relative variance change',
x = 'Participant ID')
One value (PID11) is below the cutoff for the relative variance change for random slope and intercept covariance. The extent of the deviation is minor, and was ignored.
Fitted values
Assessing whether observations are unusual with regard to the fitted values and explanatory variables using leverage. We assessed leverage at two levels: i) fixed effects, and ii) unconfounded (by fixed effects) random effects.
dotplot_diag(x = influence_leverage[, 2],
cutoff = "internal",
name = "leverage") +
labs(title = 'Leverage: fixed effects',
subtitle = 'Cutoffs determined by measures of internal scaling',
y = 'Leverage',
x = 'Participant ID')
dotplot_diag(x = influence_leverage[, 4],
cutoff = "internal",
name = "leverage") +
labs(title = 'Leverage: unconfounded random effects',
subtitle = 'Cutoffs determined by measures of internal scaling',
y = 'Leverage',
x = 'Participant ID')
Participants 5 and 12 were identified as having high leverage. Their data is highlighted in the plot below. Data look okay.
data_tm %>%
mutate(faux_colour = case_when(
PID == 'ID05' ~ 'high leverage ID05',
PID == 'ID12' ~ 'high leverage ID12',
TRUE ~ 'low leverage'
)) %>%
ggplot(data = .) +
aes(x = intensity,
y = tri_mean,
colour = faux_colour) +
geom_point(position = position_jitter(width = 0.05)) +
geom_smooth(aes(colour = faux_colour),
method = 'loess',
se = FALSE,
size = 0.6) +
scale_colour_manual(name = 'Leverage points',
values = c('#0072B2', '#D55E00', '#999999')) +
labs(title = 'Inspection of high-leverage participants',
x = 'Stimulus intensity (J)',
y = 'SPARS rating [-50 to 50]') +
scale_y_continuous(limits = c(-50, 50)) +
scale_x_continuous(breaks = seq(from = 1, to = 4, by = 0.5))
Fixed effects
Influence points were assessed by calculating Cook’s Distance metrics.
# Plot data
dotplot_diag(x = influence_cooks,
cutoff = "internal",
name = "cooks.distance") +
labs(title = 'Influence: Cooks Distance',
subtitle = 'Cutoffs determined by measures of internal scaling',
y = 'Cooks Distance',
x = 'Participant ID')
Based on There are no influential fixed effects.
Summary
The cubic model is well-specified.
Quantile mixed model regression
# Quantile model with 2.5, 25, 50, 75, and 97.5% quantiles
qmm <- lqmm(fixed = tri_mean ~ poly(intensity, 3),
random = ~ intensity,
group = PID,
data = data_tm,
tau = c(0.025, 0.25, 0.5, 0.75, 0.975))
# Summary
summary(qmm)## Call: lqmm(fixed = tri_mean ~ poly(intensity, 3), random = ~intensity,
## group = PID, tau = c(0.025, 0.25, 0.5, 0.75, 0.975), data = data_tm)
##
## tau = 0.025
##
## Fixed effects:
## Value Std. Error lower bound upper bound Pr(>|t|)
## (Intercept) -36.3724 23.1888 -82.9719 10.227 0.12319
## poly(intensity, 3)1 204.7079 19.4576 165.6065 243.809 3.633e-14
## poly(intensity, 3)2 11.5495 20.3724 -29.3903 52.489 0.57336
## poly(intensity, 3)3 26.7629 12.5693 1.5039 52.022 0.03829
##
## (Intercept)
## poly(intensity, 3)1 ***
## poly(intensity, 3)2
## poly(intensity, 3)3 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## tau = 0.25
##
## Fixed effects:
## Value Std. Error lower bound upper bound Pr(>|t|)
## (Intercept) -16.06242 7.70063 -31.53741 -0.5874 0.042220
## poly(intensity, 3)1 205.06628 21.11530 162.63351 247.4991 5.267e-13
## poly(intensity, 3)2 0.84314 10.21615 -19.68698 21.3733 0.934562
## poly(intensity, 3)3 21.92427 7.56443 6.72298 37.1256 0.005597
##
## (Intercept) *
## poly(intensity, 3)1 ***
## poly(intensity, 3)2
## poly(intensity, 3)3 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## tau = 0.5
##
## Fixed effects:
## Value Std. Error lower bound upper bound Pr(>|t|)
## (Intercept) 3.2873 8.2008 -13.1928 19.767 0.690273
## poly(intensity, 3)1 204.0394 20.6048 162.6324 245.446 2.784e-13
## poly(intensity, 3)2 2.2389 9.5865 -17.0259 21.504 0.816312
## poly(intensity, 3)3 22.1176 7.1739 7.7011 36.534 0.003361
##
## (Intercept)
## poly(intensity, 3)1 ***
## poly(intensity, 3)2
## poly(intensity, 3)3 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## tau = 0.75
##
## Fixed effects:
## Value Std. Error lower bound upper bound Pr(>|t|)
## (Intercept) 19.0218 8.0334 2.8781 35.166 0.021883
## poly(intensity, 3)1 203.2674 20.8881 161.2912 245.244 4.934e-13
## poly(intensity, 3)2 5.9630 9.6112 -13.3514 25.277 0.537853
## poly(intensity, 3)3 22.6834 7.4195 7.7733 37.594 0.003612
##
## (Intercept) *
## poly(intensity, 3)1 ***
## poly(intensity, 3)2
## poly(intensity, 3)3 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## tau = 0.975
##
## Fixed effects:
## Value Std. Error lower bound upper bound Pr(>|t|)
## (Intercept) 22.0604 7.4717 7.0455 37.075 0.004828
## poly(intensity, 3)1 188.9824 21.9912 144.7894 233.175 2.398e-11
## poly(intensity, 3)2 22.3598 13.0129 -3.7905 48.510 0.092058
## poly(intensity, 3)3 12.1005 9.1610 -6.3093 30.510 0.192681
##
## (Intercept) **
## poly(intensity, 3)1 ***
## poly(intensity, 3)2 .
## poly(intensity, 3)3
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Null model (likelihood ratio):
## [1] 132.7 (p = 0) 270.8 (p = 0) 271.1 (p = 0) 247.2 (p = 0) 188.1 (p = 0)
## AIC:
## [1] 2304 (df = 7) 1892 (df = 7) 1858 (df = 7) 1913 (df = 7) 2212 (df = 7)# Get predicted values
## Level 0 (conditional, note difference to the lmer diagnostics)
quant_predict <- as.data.frame(predict(qmm, level = 0))
names(quant_predict) <- paste0('Q', c(2.5, 25, 50, 75, 97.5))
# Join with 'central_lmm'
data_lqmm <- data_tm %>%
bind_cols(quant_predict)
# Trim prediction to upper and lower limits of the scale
data_lqmm %<>%
mutate_if(is.numeric,
funs(ifelse(. > 50,
yes = 50,
no = ifelse(. < -50,
yes = -50,
no = .))))
# Plot
ggplot(data = data_lqmm) +
aes(x = intensity,
y = Q50) +
geom_ribbon(aes(ymin = `Q2.5`,
ymax = `Q97.5`),
fill = '#E69F00') +
geom_ribbon(aes(ymin = `Q25`,
ymax = `Q75`),
fill = '#56B4E9') +
geom_point(size = 3,
shape = 21,
fill = '#FFFFFF',
colour = '#000000') +
geom_hline(yintercept = 0,
linetype = 2) +
labs(title = paste('Quantile regression'),
subtitle = 'Open circles: 50th percentile (median) | Blue band: interquartile range | Orange band: 95% prediction interval',
x = 'Stimulus intensity (J)',
y = 'SPARS rating [-50 to 50]') +
scale_y_continuous(limits = c(-50, 50)) +
scale_x_continuous(breaks = unique(data_lqmm$intensity)) 
## With original data
ggplot(data = data_lqmm) +
aes(x = intensity,
y = Q50) +
geom_ribbon(aes(ymin = `Q2.5`,
ymax = `Q97.5`),
fill = '#E69F00') +
geom_ribbon(aes(ymin = `Q25`,
ymax = `Q75`),
fill = '#56B4E9') +
geom_point(data = data_tm,
aes(y = tri_mean),
position = position_jitter(width = 0.03)) +
geom_point(size = 3,
shape = 21,
fill = '#FFFFFF',
colour = '#000000') +
geom_hline(yintercept = 0,
linetype = 2) +
labs(title = paste('Quantile regression (with original Tukey trimean data)'),
subtitle = 'Open circles: 50th percentile (median) | Blue band: interquartile range | Orange band: 95% prediction interval',
x = 'Stimulus intensity (J)',
y = 'SPARS rating [-50 to 50]') +
scale_y_continuous(limits = c(-50, 50)) +
scale_x_continuous(breaks = unique(data_lqmm$intensity)) 
There is good stability in the shape of the response characteristics across the quantiles. For all stimulus intensities, the distribution is left skewed (long tail towards lower ratings).
Session information
sessionInfo()## R version 3.4.3 (2017-11-30)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS High Sierra 10.13.3
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib
##
## locale:
## [1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8
##
## attached base packages:
## [1] methods stats graphics grDevices utils datasets base
##
## other attached packages:
## [1] bindrcpp_0.2 car_2.1-6 sjPlot_2.4.1
## [4] HLMdiag_0.3.1 lqmm_1.5.3 lme4_1.1-15
## [7] Matrix_1.2-12 patchwork_0.0.1 forcats_0.2.0
## [10] stringr_1.2.0 dplyr_0.7.4 purrr_0.2.4
## [13] readr_1.1.1 tidyr_0.8.0 tibble_1.4.2
## [16] ggplot2_2.2.1.9000 tidyverse_1.2.1 magrittr_1.5
##
## loaded via a namespace (and not attached):
## [1] TH.data_1.0-8 minqa_1.2.4 colorspace_1.3-2
## [4] modeltools_0.2-21 sjlabelled_1.0.7 rprojroot_1.3-2
## [7] estimability_1.2 snakecase_0.8.1 rstudioapi_0.7
## [10] glmmTMB_0.2.0 MatrixModels_0.4-1 DT_0.4
## [13] mvtnorm_1.0-7 lubridate_1.7.1 coin_1.2-2
## [16] xml2_1.2.0 codetools_0.2-15 splines_3.4.3
## [19] mnormt_1.5-5 knitr_1.19 sjmisc_2.7.0
## [22] effects_4.0-0 bayesplot_1.4.0 jsonlite_1.5
## [25] nloptr_1.0.4 ggeffects_0.3.1 pbkrtest_0.4-7
## [28] broom_0.4.3 shiny_1.0.5 compiler_3.4.3
## [31] httr_1.3.1 sjstats_0.14.1 emmeans_1.1
## [34] backports_1.1.2 assertthat_0.2.0 lazyeval_0.2.1
## [37] survey_3.33 cli_1.0.0 quantreg_5.34
## [40] htmltools_0.3.6 tools_3.4.3 SparseGrid_0.8.2
## [43] coda_0.19-1 gtable_0.2.0 glue_1.2.0
## [46] reshape2_1.4.3 merTools_0.3.0 Rcpp_0.12.15
## [49] carData_3.0-0 cellranger_1.1.0 nlme_3.1-131
## [52] psych_1.7.8 lmtest_0.9-35 rvest_0.3.2
## [55] mime_0.5 stringdist_0.9.4.6 MASS_7.3-48
## [58] zoo_1.8-1 scales_0.5.0.9000 hms_0.4.1
## [61] parallel_3.4.3 sandwich_2.4-0 SparseM_1.77
## [64] pwr_1.2-1 TMB_1.7.12 yaml_2.1.16
## [67] stringi_1.1.6 highr_0.6 blme_1.0-4
## [70] rlang_0.1.6 pkgconfig_2.0.1 arm_1.9-3
## [73] evaluate_0.10.1 lattice_0.20-35 prediction_0.2.0
## [76] bindr_0.1 labeling_0.3 htmlwidgets_1.0
## [79] tidyselect_0.2.3 plyr_1.8.4 R6_2.2.2
## [82] multcomp_1.4-8 RLRsim_3.1-3 withr_2.1.1.9000
## [85] pillar_1.1.0 haven_1.1.1 foreign_0.8-69
## [88] mgcv_1.8-23 survival_2.41-3 abind_1.4-5
## [91] nnet_7.3-12 modelr_0.1.1 crayon_1.3.4
## [94] rmarkdown_1.8 grid_3.4.3 readxl_1.0.0
## [97] digest_0.6.15 xtable_1.8-2 httpuv_1.3.5
## [100] stats4_3.4.3 munsell_0.4.3